Problem 51 - Algebra and Trigonometry

Question

Find the intercepts and asymptotes, and then sketch a graph of the rational function. Use a graphing device to confirm your answer. \(r(x)=\frac{x^{2}-x-6}{x^{2}+3 x}\)

Step-by-Step Solution

Answer

x-intercepts at (3,0) and (-2,0); vertical asymptotes at x=0 and x=-3; horizontal asymptote y=1.

Step 1: Find the x-intercepts

To find the x-intercepts, set the numerator of the function equal to zero: \(x^2 - x - 6 = 0\). Solve the quadratic equation using the quadratic formula \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\) where \(a = 1\), \(b = -1\), and \(c = -6\). The roots are \(x = 3\) and \(x = -2\). So, the x-intercepts are at \((3, 0)\) and \((-2, 0)\).

Step 2: Find the y-intercept

To find the y-intercept, evaluate the function at \(x = 0\): \(r(0) = \frac{0^2 - 0 - 6}{0^2 + 3 \times 0} = \frac{-6}{0}\). Therefore, the function is undefined at \(x = 0\), meaning that there is no y-intercept.

Step 3: Find the vertical asymptotes

Vertical asymptotes occur where the denominator is zero, provided the numerator is not zero at those points. Set the denominator equal to zero: \(x^2 + 3x = 0\). Factoring gives \(x(x + 3) = 0\). Solving for \(x\) gives \(x = 0\) and \(x = -3\). Therefore, vertical asymptotes are at \(x = 0\) and \(x = -3\).

Step 4: Find the horizontal asymptote

Determine the horizontal asymptote by comparing the degrees of the numerator and denominator. Since both the numerator \(x^2\) and denominator \(x^2\) have the same degree, the asymptote is the ratio of the leading coefficients, \(1/1 = 1\). Thus, the horizontal asymptote is \(y = 1\).

Step 5: Sketch the graph

Plot the intercepts, asymptotes, and determine behavior near the asymptotes. With x-intercepts at \((3, 0)\) and \((-2, 0)\), vertical asymptotes at \(x = 0\) and \(x = -3\), and a horizontal asymptote of \(y = 1\), the graph approaches the asymptotes near these points. Test points between and outside the intercepts to determine the function's behavior.

Key Concepts

Understanding x-intercepts

The x-intercepts of a rational function are points where the graph crosses the x-axis. These occur where the numerator of the function equals zero, because at these values, the output of the function is zero. For example, in the given function, the numerator is a quadratic expression: \(x^2 - x - 6\). To find the x-intercepts, this expression is set to zero: \(x^2 - x - 6 = 0\).

Factor the quadratic expression or use the quadratic formula to solve for x, resulting in points where the function equals zero. These solutions represent the x-coordinates of x-intercepts. For our function, solving gives us \(x = 3\) and \(x = -2\). Thus, the x-intercepts are at the points \((3, 0)\) and \((-2, 0)\).

When analyzing these, remember:

• Each x-intercept is a point on the graph where the function crosses or touches the x-axis.
• An x-intercept does not exist where the denominator is zero because the function would be undefined.

Understanding vertical asymptotes

Vertical asymptotes are lines that the graph of the function approaches but never touches, occurring where the denominator is zero and the numerator is not zero. For our function, the denominator is \(x^2 + 3x\).

To find where vertical asymptotes occur, set the denominator equal to zero: \(x^2 + 3x = 0\). Solving this gives \(x(x + 3) = 0\), resulting in solutions \(x = 0\) and \(x = -3\). Thus, vertical asymptotes for this function are at \(x = 0\) and \(x = -3\). These are vertical lines which the graph will infinitely approach but not cross.

Key points about vertical asymptotes:

• The graph of the function will rise or fall indefinitely as it approaches these lines.
• The behavior of the function near these lines can be examined by plugging in values just greater or less than the asymptote into the function.
• There can be no y-intercept at a vertical asymptote as the function is undefined.

Understanding horizontal asymptotes

Horizontal asymptotes are horizontal lines that indicate the end behavior of a function as \(x\) approaches positive or negative infinity. They exist based on the degrees of the numerator and the denominator in a rational function.

If the degrees are equal, the horizontal asymptote is the ratio of the leading coefficients. In our function, both the numerator and denominator have a degree of 2. The leading coefficients are both 1. Therefore, the horizontal asymptote is \(y = \frac{1}{1} = 1\).

Consider the following when dealing with horizontal asymptotes:

• As \(x\) goes to infinity or negative infinity, the graph of the function will approach the horizontal asymptote.
• Unlike vertical asymptotes, horizontal asymptotes can actually be crossed by the graph, typically near the origin if at all.
• Horizontal asymptotes provide insight into the long-term behavior of the graph.

Using the quadratic formula

The quadratic formula is a powerful tool for finding the roots of a quadratic equation, providing the x-coordinates of any x-intercepts of a graph. It is used when factoring the quadratic expression is too challenging or impossible.

For a quadratic equation in the form \(ax^2 + bx + c = 0\), the quadratic formula is given by:
\[x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\]

Applying the quadratic formula involves steps:

• Identify and assign the values of \(a\), \(b\), and \(c\) from the quadratic equation.
• Substitute these values into the quadratic formula.
• Compute the discriminant \(b^2 - 4ac\). If the discriminant is positive, there are two real roots. If zero, one real root, and if negative, no real roots.
• Solve for \(x\), obtaining the roots that give x-intercepts in rational functions.

With the expression \(x^2 - x - 6\), using \(a = 1\), \(b = -1\), \(c = -6\), the formula yields the roots \(x = 3\) and \(x = -2\), matching the earlier factor-based results.